Question

x⁴ -3x³-9x²+23x-12
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Answer 1

Respuesta:

$x^4-3x^3-9x^2+23x-12$

$\mathrm{Utilizar\:el\:teorema\:de\:la\:raiz\:racional}$

$a_0=12,\:\quad a_n=1\\\\\mathrm{Los\:divisores\:de\:}a_0:\quad 1,\:2,\:3,\:4,\:6,\:12,\:\quad \mathrm{Los\:divisores\:de\:}a_n:\quad 1\\\\\mathrm{Por\:lo\:tanto,\:verificar\:los\:siguientes\:numeros\:racionales:\quad }\pm \frac{1,\:2,\:3,\:4,\:6,\:12}{1}\\\\\frac{1}{1}\mathrm{\:es\:la\:raiz\:de\:la\:expresion,\:por\:lo\:tanto,\:factorizar\:}x-1$

$\left(x-1\right)\frac{x^4-3x^3-9x^2+23x-12}{x-1}$

$\frac{x^4-3x^3-9x^2+23x-12}{x-1}$  

$\mathrm{Dividir\:los\:coeficientes\:de\:los\:terminos\:de\:mayor\:grado\:del\:numerador\:}x^4-3x^3-9x^2+23x-12\\\\\mathrm{y\:el\:divisor\:}x-1\mathrm{\::\:}\frac{x^4}{x}=x^3$

$\mathrm{Cociente}=x^3$

$\mathrm{Multiplicar\:}x-1\mathrm{\:por\:}x^3:\:x^4-x^3\\\\\mathrm{Substraer\:}x^4-x^3\mathrm{\:de\:}x^4-3x^3-9x^2+23x-12\mathrm{\:para\:obtener\:un\:nuevo\:residuo}$

$\mathrm{Residuo}=-2x^3-9x^2+23x-12\\\\\mathrm{Por\:lo\:tanto}\\\\\frac{x^4-3x^3-9x^2+23x-12}{x-1}=x^3+\frac{-2x^3-9x^2+23x-12}{x-1}$

$\mathrm{Dividir\:los\:coeficientes\:de\:los\:terminos\:de\:mayor\:grado\:del\:numerador\:}-2x^3-9x^2+23x-12\\\\\mathrm{y\:el\:divisor\:}x-1\mathrm{\::\:}\frac{-2x^3}{x}=-2x^2\\\\\mathrm{Cociente}=-2x^2\\\\\mathrm{Multiplicar\:}x-1\mathrm{\:por\:}-2x^2:\:-2x^3+2x^2\\\\\mathrm{Substraer\:}-2x^3+2x^2\mathrm{\:de\:}-2x^3-9x^2+23x-12\mathrm{\:para\:obtener\:un\:nuevo\:residuo}\\\\\mathrm{Residuo}=-11x^2+23x-12\\\\\mathrm{Por\:lo\:tanto}$

$\frac{-2x^3-9x^2+23x-12}{x-1}=-2x^2+\frac{-11x^2+23x-12}{x-1}\\\\=x^3-2x^2+\frac{-11x^2+23x-12}{x-1}$

$\mathrm{Dividir\:los\:coeficientes\:de\:los\:terminos\:de\:mayor\:grado\:del\:numerador\:}-11x^2+23x-12\\\\\mathrm{y\:el\:divisor\:}x-1\mathrm{\::\:}\frac{-11x^2}{x}=-11x\\\\\mathrm{Cociente}=-11x\\\\\mathrm{Multiplicar\:}x-1\mathrm{\:por\:}-11x:\:-11x^2+11x\\\\\mathrm{Substraer\:}-11x^2+11x\mathrm{\:de\:}-11x^2+23x-12\mathrm{\:para\:obtener\:un\:nuevo\:residuo}\\\\\mathrm{Residuo}=12x-12\\\\\mathrm{Por\:lo\:tanto}$

$\frac{-11x^2+23x-12}{x-1}=-11x+\frac{12x-12}{x-1}\\\\=x^3-2x^2-11x+\frac{12x-12}{x-1}$

$\mathrm{Dividir\:los\:coeficientes\:de\:los\:terminos\:de\:mayor\:grado\:del\:numerador\:}12x-12\\\\\mathrm{y\:el\:divisor\:}x-1\mathrm{\::\:}\frac{12x}{x}=12\\\\\mathrm{Cociente}=12\\\\\mathrm{Multiplicar\:}x-1\mathrm{\:por\:}12:\:12x-12\\\\\mathrm{Substraer\:}12x-12\mathrm{\:de\:}12x-12\mathrm{\:para\:obtener\:un\:nuevo\:residuo}\\\\\mathrm{Residuo}=0\\\\\mathrm{Por\:lo\:tanto}$

$\frac{12x-12}{x-1}=12\\\\=x^3-2x^2-11x+12$

$\left(x-1\right)\frac{x^3-2x^2-11x+12}{x-1}$

$\frac{x^3-2x^2-11x+12}{x-1}\\\\\mathrm{Dividir\:los\:coeficientes\:de\:los\:terminos\:de\:mayor\:grado\:del\:numerador\:}x^3-2x^2-11x+12\\\\\mathrm{y\:el\:divisor\:}x-1\mathrm{\::\:}\frac{x^3}{x}=x^2\\\\\mathrm{Cociente}=x^2\\\\\mathrm{Multiplicar\:}x-1\mathrm{\:por\:}x^2:\:x^3-x^2\\\\\mathrm{Substraer\:}x^3-x^2\mathrm{\:de\:}x^3-2x^2-11x+12\mathrm{\:para\:obtener\:un\:nuevo\:residuo}\\\\\mathrm{Residuo}=-x^2-11x+12\\\\\mathrm{Por\:lo\:tanto}$

$\frac{x^3-2x^2-11x+12}{x-1}=x^2+\frac{-x^2-11x+12}{x-1}$

$\mathrm{Dividir\:los\:coeficientes\:de\:los\:terminos\:de\:mayor\:grado\:del\:numerador\:}-x^2-11x+12\\\\\mathrm{y\:el\:divisor\:}x-1\mathrm{\::\:}\frac{-x^2}{x}=-x\\\\\mathrm{Cociente}=-x\\\\\mathrm{Multiplicar\:}x-1\mathrm{\:por\:}-x:\:-x^2+x\\\\\mathrm{Substraer\:}-x^2+x\mathrm{\:de\:}-x^2-11x+12\mathrm{\:para\:obtener\:un\:nuevo\:residuo}\\\\\mathrm{Residuo}=-12x+12\\\\\mathrm{Por\:lo\:tanto}$

$\frac{-x^2-11x+12}{x-1}=-x+\frac{-12x+12}{x-1}\\\\x^2-x+\frac{-12x+12}{x-1}$

$\mathrm{Dividir\:los\:coeficientes\:de\:los\:terminos\:de\:mayor\:grado\:del\:numerador\:}-12x+12\\\\\mathrm{y\:el\:divisor\:}x-1\mathrm{\::\:}\frac{-12x}{x}=-12\\\\\mathrm{Cociente}=-12\\\\\mathrm{Multiplicar\:}x-1\mathrm{\:por\:}-12:\:-12x+12\\\\\mathrm{Substraer\:}-12x+12\mathrm{\:de\:}-12x+12\mathrm{\:para\:obtener\:un\:nuevo\:residuo}\\\\\mathrm{Residuo}=0\\\\\mathrm{Por\:lo\:tanto}$

$\frac{-12x+12}{x-1}=-12\\\\x^2-x-12$

$=\left(x^2+3x\right)+\left(-4x-12\right)\\\\x^2+3x\\\\\mathrm{Aplicar\:las\:leyes\:de\:los\:exponentes}:\quad \:a^{b+c}=a^ba^c\\\\xx+3x\\\\\mathrm{Factorizar\:el\:termino\:comun\:}x\\\\x\left(x+3\right)$

$-4x-12\\\\\mathrm{Reescribir\:}12\mathrm{\:como\:}4\cdot \:3\\\\-4x-4\cdot \:3\\\\\mathrm{Factorizar\:el\:termino\:comun\:}-4\\\\-4\left(x+3\right)$

$x\left(x+3\right)-4\left(x+3\right)\\\\\mathrm{Factorizar\:el\:termino\:comun\:}x+3\\\\\left(x+3\right)\left(x-4\right)\\\\\left(x-1\right)\left(x+3\right)\left(x-4\right)\\\\\left(x-1\right)\left(x-1\right)\left(x+3\right)\left(x-4\right)\\\\\mathrm{Simplificar}\\\\=\left(x-1\right)^2\left(x+3\right)\left(x-4\right)$